How do I UV unwrap and texture onion-like object?
You want to UV unwrap it such that you get a square/rectangular UV island. Then you can use an image texture that is setup for polar coordinates. The trick with a sphere is that you have to remove the single vertex at each pole to unrwap it properly. 


Create sphere.
Delete vertices at each pole.
Set viewport to ortho mode and view the sphere directly (front or side, no angle).
Unwrap via sphere projection. Your UV island should be square.
Create a polar coordinate image texture (features run from top to bottom).
Apply to texture using UV coordinates. I've used a Texture Mapping node to increase the X scale so that it tiles more horizontally.